A purist is a person who insists on following certain rules exactly — to the letter. If you're a language purist, it upsets you to hear someone using bad grammar.
Most purists are fans of tradition and traditional rules, always sticking to those rules themselves and often instructing other people to do the same. To be a purist is to be a kind of perfectionist — a baking purist might feel that cookies made without a certain brand of chocolate chips are vastly inferior, for example. The noun purist originally referred specifically to language use, and it comes from the Latin purus, "clean, clear, or unmixed."